Wong–Zakai approximation and support theorem for SPDEs with locally monotone coefficients

In this paper we present the Wong–Zakai approximation results for a class of nonlinear SPDEs with locally monotone coefficients and driven by multiplicative Wiener noise. This model extends the classical monotone one and includes examples like stochastic 2d Navier–Stokes equations, stochastic porous medium equations, stochastic p-Laplace equations and stochastic reaction–diffusion equations. As a corollary, our approximation results also describe the support of the distribution of solutions.     

Dynamic Picone's identity and its applications

We prove some Picone-type identities and inequalities for a class of first-order nonlinear dynamic systems and derive various weighted inequalities of Wirtinger type and Hardy type on time scales. As applications we study oscillatory and related properties of these systems including Reid's roundabout theorem on disconjugacy, Sturm's separation and comparison theorems, as well as a variational method in the oscillation theory.     

Multiscale KAM theorem for Hamiltonian systems

In this paper, we study the persistence of invariant tori in nearly integrable multiscale Hamiltonian systems with highorder degeneracy in the integrable part. Such Hamiltonian systems arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem connects closely to the stability of the systems. We introduce multiscale nondegenerate condition and multiscale Diophantine condition, comparable to the usual Diophantine condition. Using quasilinear KAM method, we prove a multiscale KAM theorem.     

Traveling wave solutions in delayed higher dimensional lattice differential systems with partial monotonicity

In this paper, we consider the existence of traveling wave solutions in delayed higher dimensional lattice differential systems with partial monotonicity. By relaxing the monotonicity of the upper solutions and allowing it greater than positive equilibrium point, we establish the existence of traveling wave solutions by means of Schauder's fixed point theorem. And then, we apply our results to delayed competition‐cooperation systems on higher dimensional lattices.     

Minimal volume invariants,topological sphere theorems and biorthogonal curvature on 4‐manifolds

The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4‐manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fundamental form is suitably bounded.     

Beurling Type Theorem on the Hilbert Space Generated by a Positive Sequence

Let H²(γ) be the Hilbert space over the bidisk D² generated by a positive sequence γ={γ_nm}_n,m ≥ 0. In this paper, we prove that the Beurling type theorem holds for the shift operator on H²(γ) with γ={γ_nm}_n,m ≥ 0 satisfying certain series of inequalities. As a corollary, we give several applications to a class of classical analytic reproducing kernel Hilbert spaces over the bidisk D².     

The Complete Biorthogonal Expansion Theorem and Its Application to a Class of Rectangular Plate Equations

In this paper, we first establish the separable $Hamiltonian$ system of rectangular cantilever thin plate bending problems by choosing proper dual vectors. Then using the characteristics of off-diagonal infinite-dimensional $Hamiltonian$ operator matrix, we derive the biorthogonal relationships of the eigenfunction systems and based on it we further obtain the complete biorthogonal expansion theorem. Finally, applying this theorem we obtain the general solutions of rectangular cantilever thin plate bending problems with two opposite edges slidingly supported.     

Ergodic theorems for queuing systems with dependent inter-arrival times

We study a G/GI/1 single-server queuing model with i.i.d. service times that are independent of a stationary process of inter-arrival times. We show that the distribution of the waiting time converges to a stationary law as time tends to infinity provided that inter-arrival times satisfy a Gärtner-Ellis type condition. A convergence rate is given and a law of large numbers established. These results provide tools for the statistical analysis of such systems, transcending the standard case with independent inter-arrival times.     

MDPs with setwise continuous transition probabilities

This paper describes the structure of optimal policies for infinite-state Markov Decision Processes with setwise continuous transition probabilities. The action sets may be noncompact. The objective criteria are either the expected total discounted and undiscounted costs or average costs per unit time. The analysis of optimality equations and inequalities is based on the optimal selection theorem for inf-compact functions introduced in this paper.     

Solvability of an inverse problem for discontinuous Sturm–Liouville operators

In this paper, we consider the Sturm–Liouville equation with the jump conditions inside the interval (0,π). The inverse problem is studied, which consists in recovering operator coefficients from two spectra, corresponding to different boundary conditions. We prove the uniqueness theorem and provide necessary and sufficient conditions for solvability of the inverse problem. We also obtain the oscillation theorem for the eigenfunctions of the considered discontinuous boundary value problem.